On 5/23/05, aaron@...749... <aaron@...749...> wrote:
I lost a week trying to wrap my head around a proper method of approximating arc length. But I now have a prototype of the smoother path blending method that Bulia suggested on IRC last weekend. It might even work. (At least it doesn't seem to raise any exceptions.)
Bulia please try it out and let me know if it does what you suggested.
Thank you, it works perfectly as I imagined it to, and gives much better results overall than the old method, with no unexplicable stretches of twists. Just what a human designer would draw as an interpolation without knowing where the nodes are. So I think we can just remove method 1 (unless someone prefers it, of course).
The only bad news is that I then tried Xara and found that it apparently uses the same algorithm 2. And I thought I invented something new :)
Now, for total happiness, the only thing missing is non-linear interpolation. Don't worry, it's easy :) Just rename the "Interpolation Time" parameter into "Exponent", with the default of 0, and use the formula
power = exponent >= 0? (1 + exponent) : 1/(1 - exponent)
effective_t = real_t ^ power
where real_t is the actual interpolation value for a given step (e.g. 1/20 for the first step of a 20-step blend), measured from 0 to 1; and effective_t is the value which you will use in the interpolating formulas instead of real_t for that step. The exponent of 0 gives linear interpolation (same as now); positive exponents shift the blend towards one of the paths, negative ones towards the other.